In the first part
of this talk I will discuss how one can characterize geometry of quantum phases
and phase transitions based on the Fubini-Study metric, which characterizes the
distance between ground state wave-functions in the external parameter space.
This metric is closely related to the Berry curvature. I will show that there
are new geometric invariants based on the Euler characteristic.

I will also show how one can directly measure this metric
tensor in simple dynamical experiments. In the second part of the talk I will
discuss emergent nature of macroscopic equations of motion (like Newton's
equations) showing that they appear in the leading order of non-adiabatic
expansion. I will show that the Berry curvature gives the Coriolis force and
the Fubini-Study metric tensor is closely related to the inertia mass. Thus I
will argue that any motion (not necessarily motion in space) is geometrical in
nature.